Unit 5 Class 9 Math Solutions – Linear Equations and Inequalities
Unit 5 Class 9 Math Solutions contain complete exercise-wise PDF solutions of the chapter Linear Equations and Inequalities. Students can find the solutions of Exercise 5.1, Exercise 5.2, and Review Exercise 5 on this page.
This unit explains linear equations, linear inequalities, real-line representations, half-planes, simultaneous inequalities, feasible regions, objective functions, and linear programming.
The uploaded PDFs include complete calculations, number-line diagrams, boundary lines, shaded solution regions, feasible-region graphs, corner-point tables, and final answers for the Punjab Board Class 9 Mathematics book.
Students can use these solutions for homework, revision, class tests, annual examinations, and board exam preparation.
Solutions of Exercise 5.1 Unit 5 Class 9 Math Notes
Exercise 5.1 of Unit 5 Class 9 Math Notes covers linear equations and inequalities. Students solve equations and inequalities, represent answers on a real line, graph inequalities in the coordinate plane, and find common solution regions of pairs of inequalities.
Solving Linear Equations
A linear equation contains a variable with highest power one. The basic aim is to isolate the variable while keeping both sides equal.
The same operation must be performed on both sides of an equation.
For example:
\(12x+30=-6\)
Subtract 30 from both sides:
\(12x=-36\)
Then divide by 12:
\(x=-3\)
Equations Containing Fractions
When an equation contains fractions, multiply every term by the least common multiple of the denominators.
For example:
\(\frac{x}{2}-\frac{3x}{4}=\frac{1}{12}\)
The least common multiple of 2, 4, and 12 is 12. Multiplying the complete equation by 12 gives:
\(6x-9x=1\)
Therefore:
\(-3x=1\)
and:
\(x=-\frac{1}{3}\)
Students should multiply every term, including terms inside brackets, by the chosen common multiple.
Opening Brackets and Combining Like Terms
When brackets are present, first apply the distributive property and then combine like terms.
For example:
\(2=7(2x+4)+12x\)
Opening the bracket gives:
\(2=14x+28+12x\)
After combining like terms:
\(2=26x+28\)
The final solution is:
\(x=-1\)
Representing an Equation on the Real Line
A linear equation in one variable usually gives one exact value. That value is shown by a filled point on the real line.
For example, the solution:
\(x=6\)
is represented by a single filled point at 6.
Solving Linear Inequalities
A linear inequality is solved in almost the same way as a linear equation.
The major difference is that the inequality sign must be reversed when both sides are multiplied or divided by a negative number.
For example:
\(-2x<8\)
Dividing by \(-2\) reverses the sign:
\(x>-4\)
Open and Closed Circles on a Number Line
A closed or filled circle is used when the endpoint is included.
The symbols that include the endpoint are:
\(\leq\)
and:
\(\geq\)
An open circle is used when the endpoint is not included.
The symbols that exclude the endpoint are:
\(<\)
and:
\(>\)
Direction of the Ray
For a solution such as:
\(x\leq4\)
the ray extends to the left because the solution contains numbers smaller than or equal to 4.
For a solution such as:
\(x\geq2\)
the ray extends to the right because the solution contains numbers greater than or equal to 2.
Interval Notation
An inequality can also be written in interval notation.
For example:
\(x<3\)
can be written as:
\(x\in(-\infty,3)\)
Similarly:
\(x\leq-1\)
can be written as:
\(x\in(-\infty,-1]\)
A round bracket shows that the endpoint is excluded, while a square bracket shows that it is included.
Graphing One Linear Inequality in Two Variables
To graph a linear inequality in the coordinate plane, first replace the inequality sign with an equality sign.
This produces the boundary line.
For example:
\(2x+y\leq6\)
has the boundary line:
\(2x+y=6\)
The line passes through:
\((0,6)\)
and:
\((3,0)\)
Testing a Point
After drawing the boundary line, test a convenient point to decide which side should be shaded.
The point:
\((0,0)\)
is commonly used when it does not lie on the boundary line.
For:
\(2x+y\leq6\)
substitution gives:
\(2(0)+0\leq6\)
This statement is true, so the side containing the origin is shaded.
Solid and Dashed Boundary Lines
A solid boundary line is used when the inequality contains equality.
Therefore, a solid line is used for:
\(\leq\)
and:
\(\geq\)
A dashed line is used for strict inequalities:
\(<\)
and:
\(>\)
The Exercise 5.1 graphs use solid lines because the inequalities include equality.
Vertical and Horizontal Boundary Lines
An inequality involving only \(x\) gives a vertical boundary line.
For example:
\(2x+1\geq0\)
simplifies to:
\(x\geq-\frac{1}{2}\)
The boundary is the vertical line:
\(x=-\frac{1}{2}\)
An inequality involving only \(y\) gives a horizontal boundary line.
For example:
\(3y-4\leq0\)
simplifies to:
\(y\leq\frac{4}{3}\)
Common Solution Region of Two Inequalities
When two inequalities are given, graph each one separately.
The required answer is the region that satisfies both inequalities at the same time.
This common shaded part is called the common solution region.
For example:
\(x+y\geq5\)
and:
\(x-y\leq1\)
can be written as:
\(y\geq5-x\)
and:
\(y\geq x-1\)
The common solution lies above both lines.
Finding the Intersection of Boundary Lines
The exact point where two boundary lines meet is found by solving their equations simultaneously.
For:
\(x+y=5\)
and:
\(x-y=1\)
adding the equations gives:
\(2x=6\)
Therefore:
\(x=3\)
and:
\(y=2\)
The intersection point is:
\((3,2)\)
Reversing the Sign While Solving for y
When an inequality is rearranged into slope-intercept form, dividing by a negative coefficient reverses the inequality sign.
For example:
\(x-2y\leq2\)
gives:
\(-2y\leq2-x\)
Dividing by \(-2\) gives:
\(y\geq\frac{x}{2}-1\)
Forgetting to reverse the sign changes the shaded region and makes the answer incorrect.
Solutions of Exercise 5.2 Unit 5 Class 9 Math Notes
Exercise 5.2 introduces Linear Programming. Students draw constraint lines, identify feasible regions, find corner points, and evaluate an objective function to obtain a maximum or minimum value.
[Embed the Exercise 5.2 PDF here]
What Is Linear Programming?
Linear programming is a method used to find the best value of a linear objective function under a set of linear constraints.
The objective may be to maximize profit, production, output, or another quantity.
It may also be to minimize cost, time, or resource use.
Objective Function
The expression that must be maximized or minimized is called the objective function.
Examples include:
\(f(x,y)=2x+5y\)
\(z=2x+3y\)
\(z=3x+y\)
The objective function is evaluated only after the feasible-region corner points have been identified.
Constraints
The inequalities that limit the possible values of \(x\) and \(y\) are called constraints.
A typical problem may include:
\(2y-x\leq8\)
\(x-y\leq4\)
\(x\geq0\)
\(y\geq0\)
The final two inequalities restrict the solution to the first quadrant.
Boundary Lines
Replace every inequality sign with an equality sign to draw its boundary line.
For example, the constraint:
\(2y-x\leq8\)
has the boundary line:
\(2y-x=8\)
Two convenient points or intercepts are used to draw each line accurately.
Feasible Region
The feasible region is the common region that satisfies every constraint.
Each constraint shades one side of its boundary line.
The overlap of all those shaded regions is the feasible region.
Only points inside or on the boundary of this region are allowed solutions.
Corner Points or Vertices
The points where the boundary lines or coordinate axes meet are called corner points or vertices.
A maximum or minimum value of a linear objective function occurs at a corner point of the feasible region.
Therefore, every corner point must be listed before the objective function is evaluated.
Finding an Intersection Point
When two boundary lines meet, solve their equations simultaneously.
For example:
\(2y-x=8\)
and:
\(x-y=4\)
give the intersection:
\((16,12)\)
This point becomes one of the corner points of the feasible region.
Corner-Point Method
The corner-point method follows four main steps.
First, draw all boundary lines and shade the feasible region.
Second, list all vertices of the feasible region.
Third, substitute every vertex into the objective function.
Fourth, compare the obtained values and select the greatest or smallest value as required.
Maximum Value
For a maximization problem, compare the objective-function value at every corner point and select the largest one.
For example, Exercise 5.2 includes:
\(f(x,y)=2x+5y\)
with the feasible-region vertices:
\((0,0),(4,0),(16,12),(0,4)\)
The greatest objective-function value is:
\(f_{\max}=92\)
at:
\((16,12)\)
Minimum Value
For a minimization problem, evaluate the objective function at every corner point and select the smallest value.
For example:
\(z=2x+y\)
has a minimum value:
\(z_{\min}=3\)
at:
\((0,3)\)
Finding Both Minimum and Maximum
Some questions ask for both the minimum and maximum values of the same objective function.
The method remains the same: evaluate the function at each corner point.
The smallest value is the minimum and the greatest value is the maximum.
For example, the exercise includes:
\(z=3x+y\)
with:
\(z_{\min}=3\text{ at }(0,3)\)
and:
\(z_{\max}=27\text{ at }(9,0)\)
Redundant Constraints
A constraint is redundant when another constraint already restricts the feasible region more strongly.
The redundant boundary line may appear on the graph without forming an edge of the final feasible region.
Students should still check that every listed corner point satisfies all constraints.
Bounded and Unbounded Feasible Regions
A bounded feasible region is enclosed on all sides and has a finite polygonal shape.
An unbounded feasible region extends indefinitely in one or more directions.
An unbounded region may still have a valid minimum or maximum, depending on the objective function.
The direction in which the objective function increases should be considered when explaining an optimum in an unbounded region.
Solutions of Review Exercise 5 Unit 5 Class 9 Math Notes
Review Exercise 5 revises the complete chapter. It includes multiple-choice questions, linear equations, inequalities on the real line, simultaneous inequalities, solution regions, and maximum and minimum values of objective functions.
[Embed the Review Exercise 5 PDF here]
Multiple-Choice Questions
The review begins with questions about linear equations, inequalities, interval notation, half-planes, associated equations, vertices, feasible regions, and objective functions.
Students should read the mathematical symbol carefully before choosing an option.
An equality sign identifies an equation, while \(<\), \(>\), \(\leq\), or \(\geq\) identifies an inequality.
Equations and Inequalities on the Real Line
The review includes equations containing fractions and inequalities with brackets.
Students must multiply by the correct common denominator, open brackets carefully, isolate the variable, and draw the correct real-line representation.
For example:
\(x<3\)
uses an open circle at 3 and a ray extending to the left.
The interval form is:
\((-\infty,3)\)
Simultaneous Linear Inequalities
The review includes pairs of inequalities whose common solution regions must be graphed.
The boundary lines are drawn first, and the required side of each line is selected.
The exact point of intersection is found by solving the boundary equations simultaneously.
For example, the lines:
\(2x+y=4\)
and:
\(x+2y=6\)
meet at:
\(\left(\frac{2}{3},\frac{8}{3}\right)\)
Maximum Value in the Review Exercise
The review includes the objective function:
\(g(x,y)=x+4y\)
subject to:
\(x+y\leq4,\quad x\geq0,\quad y\geq0\)
The feasible region has corner points:
\((0,0),(4,0),(0,4)\)
The maximum value is:
\(g_{\max}=16\)
at:
\((0,4)\)
Minimum Value in an Unbounded Region
The review also includes a minimization problem with an unbounded feasible region.
The objective function is:
\(f(x,y)=3x+5y\)
The lower boundary has the corner points:
\((0,2),\left(\frac{3}{2},\frac{1}{2}\right),(3,0)\)
The minimum value is:
\(f_{\min}=7\)
at:
\(\left(\frac{3}{2},\frac{1}{2}\right)\)
Because both coefficients in the objective function are positive, moving upward or to the right increases its value. Therefore, no point farther inside the unbounded feasible region gives a smaller result.
Important Rules of Unit 5 Class 9 Math
Balance Rule for Equations
Perform the same operation on both sides of an equation.
Negative Multiplication or Division Rule
When both sides of an inequality are multiplied or divided by a negative number, reverse the inequality sign.
Closed Endpoint
\(\leq\text{ or }\geq\Rightarrow\text{closed circle}\)
Open Endpoint
\(<\text{ or }>\Rightarrow\text{open circle}\)
Boundary-Line Rule
Replace the inequality sign with an equality sign to draw the boundary line.
Test-Point Rule
Test a convenient point, usually \((0,0)\), to determine the correct half-plane.
Common-Region Rule
For simultaneous inequalities, the answer is the region satisfying every inequality.
First-Quadrant Conditions
\(x\geq0,\qquad y\geq0\)
Objective Function
\(Z=ax+by\)
Corner-Point Principle
A maximum or minimum value of a linear objective function occurs at a corner point of the feasible region.
Intersection of Two Lines
Solve the two boundary equations simultaneously to find their exact intersection point.
Common Mistakes in Unit 5 Class 9 Math Notes
Students sometimes perform an operation on only one side of an equation.
A common mistake is forgetting to reverse an inequality sign after multiplying or dividing by a negative number.
Some students use a filled circle for a strict inequality or an open circle when the endpoint should be included.
The ray may be drawn in the wrong direction on the number line.
Students sometimes multiply only selected terms when clearing fractions instead of multiplying the complete equation.
Brackets are often opened incorrectly, especially when a negative sign appears before them.
In coordinate-plane questions, students may graph the original inequality instead of first drawing its associated boundary equation.
A test point lying on the boundary line should not be used to choose the shaded side.
Students sometimes shade the correct side of each line separately but fail to identify the common overlap.
When solving for \(y\), dividing by a negative coefficient without reversing the sign gives the wrong half-plane.
In linear programming, missing one corner point can lead to an incorrect maximum or minimum.
Some students evaluate the objective function at random interior points instead of all vertices.
A boundary intersection should be found exactly by simultaneous equations rather than estimated only from the graph.
The final answer should include both the optimum value and the point where it occurs.
Exam Preparation Tips for Unit 5 Class 9 Math Notes
Practise solving linear equations containing brackets and fractions.
Memorize when an inequality sign must be reversed.
Learn the connection between inequality symbols and open or closed circles.
Practise writing solutions in both inequality and interval forms.
Use two accurate points to draw every boundary line.
Label intercepts and intersection points clearly on graphs.
Test \((0,0)\) whenever it is not on the boundary line.
For simultaneous inequalities, shade lightly so that the common region remains visible.
In linear programming, write the corner points before evaluating the objective function.
Prepare a small table showing each corner point and its objective-function value.
Check that every vertex satisfies all constraints.
Attempt Review Exercise 5 independently before opening the solution PDF.
Why Unit 5 Class 9 Math Solutions Are Important
Unit 5 Class 9 Math Solutions are important because equations and inequalities are used throughout algebra, geometry, graphs, statistics, science, economics, and higher mathematics.
Graphing inequalities helps students understand how equations divide the coordinate plane into regions.
Linear programming introduces an important method for making the best decision when resources or conditions are limited.
A strong understanding of this unit prepares students for simultaneous equations, coordinate geometry, functions, optimization, and advanced algebra.
FAQs About Unit 5 Class 9 Math Solutions
What is the topic of Unit 5 Class 9 Math?
The topic of Unit 5 Class 9 Math is Linear Equations and Inequalities.
How many exercises are included in Unit 5?
Unit 5 includes Exercise 5.1, Exercise 5.2, and Review Exercise 5.
What is covered in Exercise 5.1?
Exercise 5.1 covers linear equations, linear inequalities, real-line representations, half-planes, and common solution regions of simultaneous inequalities.
What is covered in Exercise 5.2?
Exercise 5.2 covers linear programming, feasible regions, corner points, and maximum or minimum values of objective functions.
What is covered in Review Exercise 5?
Review Exercise 5 revises equations, inequalities, interval notation, graphs of simultaneous inequalities, feasible regions, and optimization.
When is the inequality sign reversed?
The inequality sign is reversed when both sides are multiplied or divided by a negative number.
When is a closed circle used on a number line?
A closed circle is used for \(\leq\) or \(\geq\) because the endpoint is included.
When is an open circle used on a number line?
An open circle is used for \(<\) or \(>\) because the endpoint is excluded.
How is a linear inequality graphed?
Replace the inequality sign with equality, draw the boundary line, test a point, and shade the correct half-plane.
What is a feasible region?
The feasible region is the common region that satisfies all constraints in a linear programming problem.
What is an objective function?
An objective function is the linear expression whose value must be maximized or minimized.
Where does the maximum or minimum value occur?
For the Class 9 corner-point method, the maximum or minimum value is found by evaluating the objective function at every corner point of the feasible region.
Are these Unit 5 Class 9 Math Solutions available in PDF format?
Yes. Exercise 5.1, Exercise 5.2, and Review Exercise 5 solutions are available in PDF format on this page.
Are these solutions useful for exam preparation?
Yes. Students can use these solutions for homework, revision, class tests, annual examinations, and board exam preparation.
Disclaimer
These Unit 5 Class 9 Math Solutions are prepared for educational help. Students should use them to understand the method and check their work.
Students should also study the official Class 9 Mathematics textbook and follow the method recommended by their teacher.
Final Words
Unit 5 Class 9 Math Solutions help students understand Linear Equations and Inequalities in a clear and organized way.
Exercise 5.1 develops equation-solving, number-line, and graphing skills. Exercise 5.2 applies inequalities to feasible regions and linear programming.
Study both exercises carefully, draw every graph neatly, and use Review Exercise 5 to test your understanding of the complete unit.
